Workshop

Workshop on degenerations of Calabi-Yau manifolds (IHP, Paris, May 14-16, 2018)

Valentino Tosatti, Chenyang Xu and Sébastien Boucksom are coorganizing a workshop on degenerations of Calabi-Yau manifolds at the Institut Henri Poincaré (Paris), from May 14 to 16, 2018.

Program: the workshop consists in three-hour mini-courses and one-hour talks.

Mini-courses

Sébastien Boucksom: Calabi-Yau degenerations and hybrid spaces.

Valentino Tosatti: Adiabatic limits of Ricci-flat Kähler metrics.

Chenyang Xu: Essential skeleta and dlt models.

Talks

Ruadhaí Dervan: Canonical metrics on fibrations and maps in the adiabatic limit

Yang Li: Collapsing Calabi-Yau metrics on Lefschetz K3 fibred 3-folds

Yuchen Liu: The normalized volume of a singularity is lower semi-continuous

Tony Yue Yu: The Frobenius structure conjecture in dimension two

Yuguang Zhang: Equivalences for degenerations of Calabi-Yau manifolds

Practical information and Schedule:

The meeting will take place in the room 314 of the Institut Henri Poincaré

	Monday, May 14	Tuesday, May 15	Wednesday, May 16
10.00am - 10.45am	Tosatti	Tosatti	Tosatti
10.45am - 11.15am	Coffee Break	Coffee Break	Coffee Break
11.15am - 12.00pm	Xu	Xu	Xu
12.00pm - 1.30pm	Lunch Break	Lunch Break	Lunch Break
1.30pm - 2.15pm	Boucksom	Boucksom	Boucksom
2.15pm - 2.30pm	Break	Break	Break
2.30pm - 3.30pm	Zhang	Yu	Dervan
3.30pm - 4.00pm	Coffee Break	Coffee Break	Coffee Break
4.00pm - 5.00pm	Li	Liu

References and abstracts

S.Boucksom, M.Jonsson: Tropical and non-Archimedean limits of degenerating families of volume forms.

T.de Fernex, J.Kollár, C.Xu: The dual complex of singularities

M.Gross, V.Tosatti, Y.Zhang: Collapsing of abelian fibred Calabi-Yau manifolds

J.Kollár, C.Xu: The dual complex of Calabi-Yau pairs

J.Nicaise, C.Xu: The essential skeleton of a degeneration of algebraic varieties

J.Nicaise, C.Xu, T.Yue Yu: The non-archimedean SYZ fibration

V.Tosatti: Adiabatic limits of Calabi-Yau metrics.

V.Tosatti, Y.Zhang: Collapsing hyperkähler manifolds

Ruadhaí Dervan: A natural question is when a Kähler manifold admits a canonical choice of Kähler metric. The sort of canonical metrics I will discuss are a generalisation of Kähler Einstein metrics. I will discuss some existence and non-existence results in the situation when the Kähler manifold is fibred over a lower dimensional manifold. It is worth remarking that this is a sort of inverse to Tosatti's mini-course; we attempt to understand the existence of canonical metrics on fibred manifolds through properties of the base, rather than understanding limiting behaviour of families of canonical metrics on fibred (Calabi-Yau) manifolds, which then induce certain metrics on the base. This is joint work with Julius Ross and Lars Sektnan.

Yang Li: I will discuss the problem of describing the collapsing CY metrics on a CY 3-fold with a Lefschetz K3 fibration. Collapsing CY metrics is a well studied subject, but most of the previous works concentrate on the behaviour away from the singular fibres, and the full description of the metric was only available in a very small number of cases, mostly relying on very favourable gluing ansatz.
The main geometric insight is that at the scale where the fibres have volume 1, the neighbourhood of the nodal fibre has local non-collapsing behaviour, and converges in the pointed Gromov Hausdorff sense to nodal K3 times C. Furthermore, there is a much finer scale near the nodal points in the fibration, where the scaled limit is a CY metric on C³ with maximal volume growth and singular tangent cone at infinity. This model metric was previously constructed by the author in a separate work. The difficulty of the gluing lies in the coarse nature of the gluing ansatz, and the fact that the metric has many types of characteristic behaviours at different scales. We overcome this by developing a sharp linear theory, using some earlier ideas of Gábor Székelyhidi.

Yuchen Liu: Motivated by work in differential geometry, Chi Li introduced the normalized volume of a klt singularity as the minimum normalized volume of all valuations centered at the singularity. This invariant carries some interesting geometric/topological information of the singularity. In this talk, we show that in a Q-Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. As an application, we show that K-semistability is a very generic or empty property in a Q-Gorenstein flat family of Q-Fano varieties. This is a joint work with Harold Blum.

Tony Yue Yu: The Frobenius structure conjecture is a conjecture about the geometry of rational curves in log Calabi-Yau varieties proposed by Gross-Hacking- Keel. It was motivated by the study of mirror symmetry. It predicts that the enumeration of rational curves in a log Calabi-Yau variety gives rise naturally to a Frobenius algebra satisfying nice properties. In a joint work with S. Keel, we prove the conjecture in dimension two. Our method is based on the enumeration of non-archimedean holomorphic curves developed in my thesis. We construct the structure constants of the Frobenius algebra directly from counting non-archimedean holomorphic disks. If time permits, I will also talk about compactification and extension of the algebra.

Yuguang Zhang: In this talk, we study the relationships among Ricci-flat Kähler-Einstein metrics, cohomology classes of holomorphic volume forms, and the Weil-Peterson metric of degenerations of Calabi-Yau manifolds.